Quasi-polar spaces

TL;DR

This paper explores the concept of quasi-polar spaces, introducing a generalized switching technique, analyzing its effects on polar spaces, and providing constructions and conditions for their existence, especially in specific cases.

Contribution

It introduces a generalized notion of pivoting called switching, extends it to Hermitian polar spaces, and analyzes its implications for constructing quasi-polar spaces.

Findings

Switching is the only modification for q >= 4 that creates quasi-polar spaces from hyperplanes.

Special cases for q=2 and q=3 are thoroughly investigated.

Conditions for the existence of a nucleus in parabolic quadrics in even characteristic are established.

Abstract

Quasi-polar spaces are sets of points having the same intersection numbers with respect to hyperplanes as classical polar spaces. Non-classical examples of quasi-quadrics have been constructed using a technique called pivoting [5]. We introduce a more general notion of pivoting, called switching, and also extend this notion to Hermitian polar spaces. The main result of this paper studies the switching technique in detail by showing that, for q >= 4, if we modify the points of a hyperplane of a polar space to create a quasi-polar space, the only thing that can be done is pivoting. The cases q = 2 and q = 3 play a special role for parabolic quadrics and are investigated in detail. Furthermore, we give a construction for quasi-polar spaces obtained from pivoting multiple times. Finally, we focus on the case of parabolic quadrics in even characteristic and determine under which hypotheses the existence of a nucleus (which was included in the definition given in [5]) is guaranteed.